Question 3.T.8: If (xn) is convergent, then it is a Cauchy sequence.

Suppose lim x_{n} = x and ε is any positive number. We know there is an N ∈ \mathbb{N} such that

|x_{n} − x| < ε/2  for all n ≥ N.

Now if we take m, n ≥ N then

|x_{n} − x| < ε/2, |x_{m} − x| < ε/2.

Using the triangle inequality (Theorem 2.1), we obtain

|x_{n} − x_{m}| ≤ |x_{n} − x| + |x_{m} − x| < ε,

which means (x_{n}) is a Cauchy sequence.